The number of circles that touch all the straight lines `x+y-4=0, x-y+2=0` and `y=2`, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem of finding the number of circles that touch all the given straight lines \(x+y-4=0\), \(x-y+2=0\), and \(y=2\), we can follow these steps:

### Step 1: Identify the lines and their intersections
1. **Line 1**: \(x + y - 4 = 0\) can be rewritten as \(y = 4 - x\).
2. **Line 2**: \(x - y + 2 = 0\) can be rewritten as \(y = x + 2\).
3. **Line 3**: \(y = 2\) is a horizontal line.

### Step 2: Find the points of intersection
- To find the intersection of Line 1 and Line 2:
\[ 4 - x = x + 2 \implies 4 - 2 = 2x \implies 2 = 2x \implies x = 1 \]
Substituting \(x = 1\) into Line 1:
\[ y = 4 - 1 = 3 \]
So, the intersection point is \((1, 3)\).

- To find the intersection of Line 1 and Line 3:
\[ 2 = 4 - x \implies x = 2 \]
So, the intersection point is \((2, 2)\).

- To find the intersection of Line 2 and Line 3:
\[ 2 = x + 2 \implies x = 0 \]
So, the intersection point is \((0, 2)\).

### Step 3: Visualize the lines and intersections
Now, we can visualize the lines and their intersections on a coordinate plane:
- Line 1 intersects the y-axis at (0, 4) and the x-axis at (4, 0).
- Line 2 intersects the y-axis at (0, 2) and the x-axis at (-2, 0).
- Line 3 is a horizontal line at \(y = 2\).

### Step 4: Determine the regions formed by the lines
The lines create several regions in the coordinate plane. We need to find the regions where circles can be drawn that touch all three lines.

### Step 5: Analyze the possible circles
1. **Above Line 1 and Line 3**: A circle can be drawn in this region.
2. **Between Line 1 and Line 3**: A circle can be drawn in this region.
3. **Between Line 2 and Line 3**: A circle can be drawn in this region.
4. **Below Line 2 and Line 3**: A circle can be drawn in this region.

### Conclusion
After analyzing the regions, we find that there are a total of **four distinct regions** where circles can be drawn that touch all three lines.

Thus, the number of circles that touch all the straight lines is **4**.